### Re: Factorization polynomially reducible to discrete log - known fact or not?

```
On 7/9/06, Ondrej Mikle [EMAIL PROTECTED] wrote:

I believe I have the proof that factorization of N=p*q (p, q prime) is
polynomially reducible to discrete logarithm problem. Is it a known fact
or not? I searched for such proof, but only found that the two problems
are believed to be equivalent (i.e. no proof).

Take a look at this paper: http://portal.acm.org/citation.cfm?id=894497

Eric Bach  Discrete Logarithms and Factoring

ABSTRACT: This note discusses the relationship between the two
problems of the title. We present probabilistic polynomial-time
reduction that show: 1) To factor n, it suffices to be able to compute
discrete logarithms modulo n. 2) To compute a discrete logarithm
modulo a prime power p^E, it suffices to know It mod p. 3) To compute
a discrete logarithm modulo any n, it suffices to be able to factor
and compute discrete logarithms modulo primes. To summarize: solving
the discrete logarithm problem for a composite modulus is exactly as
hard as factoring and solving it modulo primes.

Max

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### Re: Factorization polynomially reducible to discrete log - known fact or not?

```
The algorithm is very simple:
1. Choose a big random value x from some very broad range
(say, {1,2,..,N^2}).
2. Pick a random element g (mod N).
3. Compute y = g^x (mod N).
4. Ask for the discrete log of y to the base g, and get back some
answer x' such that y = g^x' (mod N).
5. Compute x-x'.  Note that x-x' is a multiple of phi(N), and
it is highly likely that x-x' is non-zero.  It is well-known
that given a non-zero multiple of phi(N), you can factor N in
polynomial time.

Not exactly. Consider N = 3*7 = 21, phi(N) = 12, g = 4, x = 2, x' = 5.
You'll only get a multiple of phi(N) if g was a generator of the
multiplicative group Z_N^*.

Peter

--
[Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI)  [ICQ] 134813278

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### Re: Factorization polynomially reducible to discrete log - known fact or not?

```
Charlie Kaufman wrote:

I believe this has been known for a long time, though I have never seen the
proof. I could imagine constructing one based on quadratic sieve.

I believe that a proof that the discrete log problem is polynomially reducible
to the factorization problem is much harder and more recent (as in sometime in
the last 20 years). I've never seen that proof either.

--Charlie

OK, I had the proof checked. I put it here:
http://www.ms.mff.cuni.cz/~miklo1am/Factorization_to_DLog.pdf

Warning: it may be not what you'd expect.

First of all, it reduces the factorization to a discrete log in a group
of unknown order (or put in another words: you'd need to factorize to
learn the group order). It has been proven by V. Shoup that when group
operation and the inverse are the only operations that can be done with
group elements, then the best algorithm can be O(sqrt(n)), where n is
the number of elements. I guess then the group of Z_N* (where N=pq) of
unknown order qualifies for this if we don't want to use factorization
(actually you can't compute inverse group operation here). In the light
of this fact, is this proof of any use?

Even if the proof is not useful, is the generator picking lemma (lemma
2) anything new? It states basically this:
In any cyclic group of order n there is at least 1/log2(n) probability
of picking a generator randomly and thus generator can be found in
polynomial time with overwhelming probability of success.

The only facts close to this lemma I found were:
1) Product phi(p_i)/p_i for consecutive primes p_i approaches zero as
more and factors are added to the product (phi is Euler phi function).
The lemma states a lower bound for the product.
2) If the generalized Riemann hypothesis is true, then for every prime
number p, there exists a primitive root modulo p that is less than 70
(ln(p))^2. (http://en.wikipedia.org/wiki/Primitive_root_modulo_n)

Charlie:
Thanks for answering my second question which I have not asked yet :-)
(the reduction in opposite direction). I'm also working on the opposite
reduction, but I'm at best halfway through (and not sure if I am able to
finish it).

Last question:
Joseph Ashwood mentioned someone who claimed to have algorithm for
factorization and had only the reduction to DLP. Anyone knows where I
could find the algorithm? Or maybe name of the person, so I could search
the web.

Thanks
O. Mikle

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### RE: Factorization polynomially reducible to discrete log - known fact or not?

```I believe this has been known for a long time, though I have never seen the
proof. I could imagine constructing one based on quadratic sieve.

I believe that a proof that the discrete log problem is polynomially reducible
to the factorization problem is much harder and more recent (as in sometime in
the last 20 years). I've never seen that proof either.

--Charlie

-Original Message-
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of Ondrej Mikle
Sent: Sunday, July 09, 2006 12:22 PM
To: cryptography@metzdowd.com
Subject: Factorization polynomially reducible to discrete log - known fact or
not?

Hello.

I believe I have the proof that factorization of N=p*q (p, q prime) is
polynomially reducible to discrete logarithm problem. Is it a known fact
or not? I searched for such proof, but only found that the two problems
are believed to be equivalent (i.e. no proof).

I still might have some error in the proof, so it needs to be checked by
someone yet. I'd like to know if it is already known (in that case there
would be no reason to bother with it).

Thanks
O. Mikle

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